QUANTUM DOUBLE-TORUS

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1 1 QUANTUM DOUBLE-TORUS arxiv:math/980309v [math.qa] 4 Sep 1998 Piotr M. Hajac Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver St., Cambridge CB3 9EW, England. pmh33@amtp.cam.ac.uk Tetsuya Masuda Institute of Mathematics, University of Tsukuba, Ibaraki, 305 Japan. tetsuya@math.tsukuba.ac.jp Abstract. A symmetry extending the T -symmetry of the noncommutative torus Tq is studied in the category of quantum groups. This extended symmetry is given by the quantum double-torus DTq defined as a compact matrix quantum group consisting of the disjoint union of T and Tq. The bicross-product structure of the polynomial Hopf algebra A(DTq ) is computed. The Haar measure and the complete list of unitary irreducible representations of DTq are determined explicitly. Résumé. Une symétrie qui prolonge la T -symétrie d un tore noncommutatif Tq est étudiée dans le contexte des groupes quantiques. Cette symétrie est donnée par le double-tore quantique DTq défini comme un groupe quantique compact matriciel, union disjointe de T et Tq. La structure de bi-produit croisée de l algèbre de Hopf polynomiale A(DTq ) est calculée. La mesure d Haar et la liste complète des représentations irreductibles unitaires de DTq sont determinées explicitement. Version française abrégée Nous prolongeons la symétrie classique T du deux-tore noncommutatif Tq à la symétrie du groupe quantique donné par le double-tore quantique DTq. Contrairement à la symétrie T de Tq, la symétrie DTq découvre la nature quantique des deux-tores noncommutatifs. Un double-tore quantique est un sous-groupe quantique (non connexe) de la q-déformation pure tordue de U() donnée par la R-matrice diagonal diag(1, q 1, q, 1). Du point de vue de la théorie de Hopf-Galois, l anneau de coordonnées de DTq est un bi-produit croisé de l algèbre d Hopf polynomiale de Z et de celle de T avec un cocycle non-trivial et une coaction qui déterminent les structures d algèbre et de cogébre, respectivement. (Le groupe quantique DTq peut être vu comme une quantification du produit semi-direct du groupe T Z.) La completion de l anneau de coordonnées de DTq nous donne une C -algèbre qui est isomorphe à la somme directe de l algèbre C(T ) des fonctions continues sur le deux-tore classique et la C -algébre C(T q ) de deux-tore noncommutatif. DTq est un groupe quantique compact matriciel dans le sens de S. L. Woronowicz. Son dual unitaire (collection des toutes les représentations irreductibles unitaires) est donné par la formule: DT q = Z Z (Z N + ). Dedicated to the memory of Kazimiera Rzeczkowska. On leave from: Department of Mathematical Methods in Physics, Warsaw University, ul. Hoża 74, Warsaw, Poland. ang.html ( pmh@fuw.edu.pl) Partially supported by the IHES visiting stipend, NATO postdoctoral fellowship and KBN grant P03A

2 Introduction The aim of this note is to construct and analyse a q-deformation of the semi-direct product group T Z that extends the T -symmetry of the noncommutative torus Tq. This semi-direct product is a subgroup of U() consisting of diagonal and off-diagonal matrices with unitary entries. Since it is a disjoint union of two tori, we call it a double torus, and denote DT. To carry out a q-deformation of DT it is convenient to start with the well-known twoparameter quantisation GL p,q (), p, q C, of GL() given by the R-matrix q (pq) 1 p 1 0. (0.1) For p = q we can consider the involutive structure given on generators by a = S(a), b = S(c), c = S(b), d = S(d), where S is the antipode. The introduction of such an involutive structure makes the quantum determinant D := ad qbc = da q 1 bc a unitary element, and defines the coordinate ring A(U q,q ()) of the compact matrix quantum group U q,q (). (See [9] for details.) For q R, the quantum determinant D becomes central, we can set it to 1, and decompose U q,q () as follows: U q,q () =: U q () = T 1 SU q (). The other characteristic case is when q U(1). Then the R-matrix (0.1) becomes diagonal, and the commutation relations between generators of A(U q 1,q()) are ab = q 1 ba, cd = q 1 dc, ac = qca, bd = qdb, ad = da, bc = q cb. This q-deformation of U() is precisely what we use to quantise the double-torus subgroup of U(). Although it is not technically needed, we assume the coefficient field to be C and introduce the involutive structure from the very beginning for the sake of brevity and clarity. For the coaction and coproduct we use the Sweedler notation with the suppressed summation sign. 1 Quantum Double-Torus as a Symmetry of T q One can represent the -torus T as the quotient R /Z or as a subset of C. The latter point of view leads to the question of which subgroups of GL(, C) preserve T. A natural subgroup of GL(, C) that preserves T is the double torus DT described in the Introduction. The double torus is a subgroup of U(), which can be viewed as a field of T 3 over the internal points of [0, 1] bounded by T at each of the endpoints. Although this geometrical picture disregards the Lie group structure of U(), it helps us to describe the support of the Haar functional on the Hopf algebra A(U()) of the polynomial functions on U(). This support is included in the space of polynomials on the aforementioned interval [0, 1], and the Haar functional is given by integration on [0, 1]. Moreover, this picture of U() allows us to locate DT inside U() as the two T placed at the endpoints of [0, 1]. What we study here is a q-deformed version of this setting. The standard q-deformation of U() turns the interval [0, 1] into the quantum interval [0, 1] q consisting of powers of q. Missing is the endpoint corresponding to the off-diagonal torus of DT. The off-diagonal part of the double-torus subgroup of U() is destroyed by this quantisation. On the other hand, the pure-twist q-deformation of U(), with q U(1), leaves the interval [0, 1] unchanged. It deforms U() into U q 1,q(), which can be understood as a field of noncommutative 3-tori over the open interval (0, 1) bounded by T q at 0 and T at 1 (see [11,

3 3 Section 6], cf. [8]). The union of T q and T is the quantum double-torus subgroup of U q 1,q(). More precisely, we have: Definition 1.1 Let I be the Hopf ideal of A(U q 1,q()) generated by {ab, ac, cd, bd}. We call the quotient Hopf algebra A(DT q ) := A(U q 1,q())/I the coordinate ring of the quantum doubletorus DT q. A(DTq ) is a -Hopf algebra. The Diamond Lemma [, Theorem 1.] enables us to claim the following: Lemma 1. Put z = D 1 ad. The set of monomials: D k z l a m c n, m > 0, n > 0; D k z l a m b n, m > 0, n < 0; D k z l d m c n, m < 0, n > 0; D k z l d m b n, m < 0, n < 0; D k z l c n, n > 0; D k z l b n, n < 0; D k z l a m, m > 0; D k z l d m, m < 0; D k z l ; k, l, m, n Z, (1.) is a C-basis of A(U q 1,q()). Denote the generators of the quotient algebra A(DTq ) by the same letters as generators of A(U q 1,q()). The set of polynomials D m a n, D m d n, D m z, D m b n, D m c n, D m (1 z); m, n Z, n > 0; (1.3) is a C-basis of A(DT q ). Remark 1.3 It is tempting to define a disconnected quantum space by requiring its function algebra to contain a non-trivial central idempotent. Since z is central and becomes an idempotent after passing to the quotient algebra A(DTq ), the quantum double-torus is disconnected according to such a definition. The main point of this section is that DT q acts the same way on T q as DT on T. Thus we obtain a quantum symmetry of T q that extends the well-studied classical T -symmetry of T q. More precisely, we have: Proposition 1.4 Let ρ c and ρ q be the appropriate left coactions induced by the standard left coaction given on generators by ( ) ( ) ( ) x a 1 b 1 1 x y c 1 d 1 1 y. Let π be the canonical projection setting z = 1. Then the following diagram is commutative: A(T q ) id A(T q ) ρ c A(T ) A(T q ) π id ρ q A(DT q ) A(T q ). A standard reasoning allows us to claim the same at the C -algebraic level. The classical T -symmetry of Tq is insensitive to the noncommutative nature of Tq. In contrast, the DTq -quantum symmetry of T q detects the square of the deformation parameter q, i.e., we have bc = q cb in A(DTq ).

4 4 Bicross-product Structure of A(DT q ) The concept of short exact sequences of groups can be extended to the category of quantum groups, where it is described by (strictly) exact sequences of Hopf algebras (see [10, 1, 1]). Consider the sequence of Hopf algebras A(Z ) i A(DTq ) π A(T ), where i is the Hopf algebra injection sending f A(Z ), f(0) := 1, f(1) := 0, to z, and π is obtained from the canonical surjection setting z = 1. It can be directly verified that this is a strictly exact sequence of Hopf algebras in the sense of [1, p.3338]. Thus we can view DTq as a quantum-group extension of Z by T. In the language of the Hopf-Galois theory, we can describe A(DTq ) as a cleft A(T )-Galois extension of A(Z ), or a crossed product algebra of A(Z ) by A(T ) (e.g., see [13]). Recall that a cleaving map of a cleft H-Galois extension B P is a convolution invertible colinear map from a Hopf algebra H to an H-comodule algebra P. In the case of A(DTq ) we can construct a cleaving map j : A(T ) A(DTq ) in the following way: D l a k l + ( 1) l q l D l c k l for k > l j(u k v l ) := D k z + ( D) k (1 z) for k = l ( 1) k q k D k b l k + D k d l k for k < l. Here u, v are generators of A(T ), and k, l Z. Observe that for q 1 the map j is a -homomorphism but not an algebra map. It is an algebra homomorphism only for q = 1. Then it can be obtained as the pull-back of the appropriate trivialisation map for the trivial principal bundle DT (Z, T ). We can think of DTq as a quantum principal bundle with the classical base space Z and classical structure group T, but whose total space consists of two non-equivalent fibres: T and T q. It is known that cleft Hopf-Galois extensions and crossed products are equivalent notions (see Theorem 1.1 in [13], [4, 6]). The crossed-product structure of A(DTq ) is given by a nontrivial cocycle σ : A(T ) A(T ) A(Z ) and the trivial action of A(T ) on A(Z ). One can compute σ according to the formula (cf. [5, Lemma 1.9]): σ(h g) = j(h (1) )j(g (1) )j 1 (h () g () ). Explicitly, we obtain: (.4) Proposition.1 The cocycle σ associated to the cleaving map (.4) is given by the formulas: σ = σ c z + σ q (1 z), σ c = ε ε (ε is the counit map), σ q (u k v l u m v n ) = q kn q m(k+m) q n(k+m) q (k+m)(l k m) q l(k+m) q k(k+n) for k > l, m > n for k > l, m = n for k > l, m < n, k + m > l + n for k > l, m < n, k + m = l + n for k > l, m < n, k + m < l + n for k = l, m > n 1 for k = l, m = n q k(k+m) for k = l, m < n q k(l+n) for k < l, m > n, k + m > l + n q (l+n)(l+n k) for k < l, m > n, k + m = l + n q m(l+n) for k < l, m > n, k + m < l + n q m(m+l) for k < l, m = n q lm for k < l, m < n.

5 5 The cocycle σ determines an algebra structure on A(Z ) A(T ) by the following rule (see [3, p.693]): (x h)(y g) = xyσ(h (1) g (1) ) h () g (). The tensor product A(Z ) A(T ) with the above algebra structure is isomorphic as an algebra to A(DT q ). One can determine the crossed coproduct structure of A(DT q ) in a similar fashion. First, we can transform a cleaving map j into a cocleaving map l given by the formula l(p) = p (0) j 1 (p (1) ). (The inverse transform is given by [1, 3..13()].) The mapping l associated to (.4) can be computed directly to be: l(d m a n ) = z, l(d m d n ) = z, l(d m z) = z, l(d m (1 z)) = ( 1) m (1 z), l(d m b n ) = ( 1) m q m (1 z), l(d m c n ) = ( 1) m q m (1 z). (.5) Here m, n Z, n > 0. It can be verified that l is a -homomorphism, and that it coincides with its convolution inverse. The crossed coproduct structure of A(DT q ) is given by a non-trivial coaction λ : A(T ) A(T ) A(Z ), and the trivial cococycle. We calculate λ from l according to the formula [1, Proposition 3..9]: λ(π(p)) = π(p () ) l 1 (p (1) )l(p (3) ). Hence we obtain: Proposition. The coaction λ associated to the cocleaving map (.5) is a -algebra homomorphism defined by λ(u m v n ) = u m v n z + u n v m (1 z). For q = 1, the coaction λ is the pull-back of the Z action on T defining DT as the semi-direct product. The fact that λ remains unchanged under the q-deformation reflects the property that matrix quantum groups, in particular DT q, have the classical (matrix) coproduct. The coaction λ determines a coalgebra structure on A(Z ) A(T ) by the following rule (see [7, (6)]): (x h) = (x (1) h (1) (0) ) (x () h (1) (1) h () ), where λ(h) := h (0) h (1). The tensor product A(Z ) A(T ) with the above coalgebra structure is isomorphic as a coalgebra to A(DT q ). The crossed-product and crossed-coproduct structures described above together form a bicross-product (see [7, p.44] for this type of bicross-products): Proposition.3 A(DT q ) is isomorphic as a Hopf algebra to the bicross-product A(Z ) λ # σ A(T ) defined by cocycle σ and coaction λ of propositions.1 and., respectively. The same kind of bicross-product structure (i.e., with the trivial action and cococycle) is studied for the Borel quantum subgroup of SL e πi/3() in [5], and affine Hopf algebra U q ( sl ˆ ) in [7]. To end this section, let us mention that for any q U(1) satisfying ( q) n = 1, n Z, n > 0, we can construct a finite dimensional Hopf algebra, by dividing A(DTq ) by the Hopf ideal generated by: a n z, b n (1 z), c n (1 z), d n z, D n 1. 3 Representation Theory of DT q Since the quantum double-torus DTq is a compact quantum group, all of the unitary irreducible representations are finite dimensional. They are obtained by the irreducible decompositions of the (multiple) tensor products of some basic representations. The obvious representation is the unitary group-like element given by the (non-central) quantum determinant D. Other basic representations are given by the (central) unitary group-like element z 1, and the natural vector representation ( ) a c b d. There exist the following three families of unitary irreducible representations: ( ) D χ m := D m, χ z m := D m (z 1), w m,n := m a n D m b n D m c n D m d n, m, n Z, n > 0. (3.6)

6 6 The uniquely determined Haar functional h on A(DTq ) is positive and faithful. It vanishes on all elements of the C-basis (1.3) of A(DTq ) except for the central idempotents z and (1 z), corresponding to the classical and quantum component of A(DTq ), respectively. (It annihilates all polynomials of χ m, χ z m and w m,n.) Let {e c m,n, e q m,n} m,n Z be an orthonormal basis of Hilbert space H. The GNS construction yields the following faithful -representation π of A(DTq ) on H: π(a)e c m,n = { e c m,n+1 for n 0 e c m+1,n+1 for n < 0, π(c)e q m,n = { e q m,n+1 for n 0 q m 1 e q m+1,n+1 for n < 0, π(b)e q m,n = { q n 1 e q m+1,n 1 for n > 0 q m e q m,n 1 for n 0, π(d)e c m,n = { e c m+1,n 1 for n > 0 e c m,n 1 for n 0, π(a)e q m,n = 0, π(b)ec m,n = 0, π(c)ec m,n = 0, π(d)eq m,n = 0. The operator norm completion C(DTq ) of A(DTq ) is a C -algebra isomorphic to the direct sum of the algebra C(T ) of continuous functions on T and the noncommutative C -algebra C(T q ), q U(1), of the quantum two-torus. This means, in particular, that the C -algebra C(DTq ) is not of type I for a generic q U(1). Standard reasoning allows us to conclude that the comultiplication extends from A(DTq ) to C(DTq ), and we have: Proposition 3.1 DTq is a compact matrix quantum group in the sense of [14, Definition 1.1]. Theorem 3. The unitary dual of the quantum double-torus is given by DT q = Z Z (Z N + ). The complete list of all unitary irreducible representations of DTq is given by the three families of representations (3.6). References [1] Andruskiewitsch N., Devoto J., Extensions of Hopf algebras, Algebra i Analiz 7 (1995) -61. [] Bergman G.M., The diamond lemma for ring theory, Adv. Math. 9 (1978) [3] Blattner R.J., Cohen M., Montgomery S., Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 98 (1986) [4] Blattner R.J., Montgomery S., Crossed products and Galois extensions of Hopf algebras, Pacific J. Math. 137 (1989) [5] D abrowski L, Hajac P.M., Siniscalco P., Explicit Hopf-Galois description of SL e πi/3()-induced Frobenius homomorphisms, q-alg/ [6] Doi Y., Takeuchi M., Cleft comodule algebras for a bialgebra, Comm. Alg. 14 (1986) [7] Majid S., Bicross-product Structure of affine quantum groups, Lett. Math. Phys. 39 (1997) [8] Matsumoto K., Non-commutative 3-spheres, In: Araki H., Choda H., Nakagami Y., Saitô K., Tomiyama J. (Eds.), Current Topics in Operator Algebras, World Scientific, 1991, pp ; Noncommutative three dimensional spheres, Japan. J. Math. 17 (1991) [9] Müller-Hoissen F., Reuten C., Bicovariant differential calculi on GL p,q () and quantum subgroups, J. Phys. A 6 (1993) [10] Parshall B., Wang J., Quantum linear groups, American Mathematical Society Memoirs no.439, 1991, Providence, R.I., American Mathematical Society. [11] Rieffel M.A., Compact quantum groups associated with toral subgroups, Contemporary Math. 145 (1993) [1] Schneider H.J., Some remarks on exact sequences of quantum groups, Comm. Alg. 1 (1993) [13] Schneider H.J., Hopf Galois extensions, crossed products, and Clifford theory, In: Bergen J., Montgomery S. (Eds.), Advances in Hopf Algebras, Lecture Notes in Pure and Applied Mathematics 158, Marcel Dekker, Inc., 1994, pp [14] Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987)